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Register a new userDistribution of spectral linear statistics on random matrices beyond the large deviation function -- Wigner time delay in multichannel disordered wires
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An invariant ensemble of $Ntimes N$ random matrices can be characterised by a joint distribution for eigenvalues $P(lambda_1,cdots,lambda_N)$. The study of the distribution of linear statistics, i.e. of quantities of the form $L=(1/N)sum_if(lambda_i)$ where $f(x)$ is a given function, appears in many physical problems. In the $Ntoinfty$ limit, $L$ scales as $Lsim N^eta$, where the scaling exponent $eta$ depends on the ensemble and the function $f$. Its distribution can be written under the form $P_N(s=N^{-eta},L)simeq A_{beta,N}(s),expbig{-(beta N^2/2),Phi(s)big}$, where $betain{1,,2,,4}$ is the Dyson index. The Coulomb gas technique naturally provides the large deviation function $Phi(s)$, which can be efficiently obtained thanks to a thermodynamic identity introduced earlier. We conjecture the pre-exponential function $A_{beta,N}(s)$. We check our conjecture on several well controlled cases within the Laguerre and the Jacobi ensembles. Then we apply our main result to a situation where the large deviation function has no minimum (and $L$ has infinite moments)~: this arises in the statistical analysis of the Wigner time delay for semi-infinite multichannel disordered wires (Laguerre ensemble). The statistical analysis of the Wigner time delay then crucially depends on the pre-exponential function $A_{beta,N}(s)$, which ensures the decay of the distribution for large argument.
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Invariant ensembles of random matrices are characterized by the distribution of their eigenvalues ${lambda_1,cdots,lambda_N}$. We study the distribution of truncated linear statistics of the form $tilde{L}=sum_{i=1}^p f(lambda_i)$ with $p<N$. This problem has been considered by us in a previous paper when the $p$ eigenvalues are further constrained to be the largest ones (or the smallest). In this second paper we consider the same problem without this restriction which leads to a rather different analysis. We introduce a new ensemble which is related, but not equivalent, to the thinned ensembles introduced by Bohigas and Pato. This question is motivated by the study of partial sums of proper time delays in chaotic quantum dots, which are characteristic times of the scattering process. Using the Coulomb gas technique, we derive the large deviation function for $tilde{L}$. Large deviations of linear statistics $L=sum_{i=1}^N f(lambda_i)$ are usually dominated by the energy of the Coulomb gas, which scales as $sim N^2$, implying that the relative fluctuations are of order $1/N$. For the truncated linear statistics considered here, there is a whole region (including the typical fluctuations region), where the energy of the Coulomb gas is frozen and the large deviation function is purely controlled by an entropic effect. Because the entropy scales as $sim N$, the relative fluctuations are of order $1/sqrt{N}$. Our analysis relies on the mapping on a problem of $p$ fictitious non-interacting fermions in $N$ energy levels, which can exhibit both positive and negative effective (absolute) temperatures. We determine the large deviation function characterizing the distribution of the truncated linear statistics, and show that, for the case considered here ($f(lambda)=1/lambda$), the corresponding phase diagram is separated in three different phases.
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The exact statistics of an arbitrary quantum observable is analytically obtained. Due to the probabilistic nature of a sequence of intermediate measurements and stochastic fluctuations induced by the interaction with the environment, the measurement outcomes at the end of the systems evolution are random variables. Here, we provide the exact large-deviation form of their probability distribution, which is given by an exponentially decaying profile in the number of measurements. The most probable distribution of the measurement outcomes in a single realization of the system transformation is then derived, thus achieving predictions beyond the expectation value. The theoretical results are confirmed by numerical simulations of an experimentally reproducible two-level system with stochastic Hamiltonian.
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